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Random Walk in Python

Learn how to use Python to make a Random Walk

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Imports

The tutorial below imports NumPy, Pandas, SciPy, and Random.

In [1]:
import plotly.plotly as py
import plotly.graph_objs as go
from plotly.tools import FigureFactory as FF

import numpy as np
import pandas as pd
import scipy
import random

Tips

A random walk can be thought of as a random process in which a tolken or a marker is randomly moved around some space, that is, a space with a metric used to compute distance. It is more commonly conceptualized in one dimension ($\mathbb{Z}$), two dimensions ($\mathbb{Z}^2$) or three dimensions ($\mathbb{Z}^3$) in Cartesian space, where $\mathbb{Z}$ represents the set of integers. In the visualizations below, we will be using scatter plots as well as a colorscale to denote the time sequence of the walk.

Random Walk in 1D

The jitter in the data points along the x and y axes are meant to illuminate where the points are being drawn and what the tendancy of the random walk is.

In [2]:
x = [0]

for j in range(100):
    step_x = random.randint(0,1)
    if step_x == 1:
        x.append(x[j] + 1 + 0.05*np.random.normal())
    else:
        x.append(x[j] - 1 + 0.05*np.random.normal())
        
y = [0.05*np.random.normal() for j in range(len(x))]
        
trace1 = go.Scatter(
    x=x,
    y=y,
    mode='markers',
    name='Random Walk in 1D',
    marker=dict(
        color=[i for i in range(len(x))],
        size=7,
        colorscale=[[0, 'rgb(178,10,28)'], [0.50, 'rgb(245,160,105)'],
                    [0.66, 'rgb(245,195,157)'], [1, 'rgb(220,220,220)']],
        showscale=True,
    )
)

layout = go.Layout(
    yaxis=dict(
        range=[-1, 1]
    )
)

data = [trace1]
fig= go.Figure(data=data, layout=layout)
py.iplot(fig, filename='random-walk-1d')
Out[2]:

Random Walk in 2D

In [3]:
x = [0]
y = [0]

for j in range(1000):
    step_x = random.randint(0,1)
    if step_x == 1:
        x.append(x[j] + 1 + np.random.normal())
    else:
        x.append(x[j] - 1 + np.random.normal())
    
    step_y = random.randint(0,1)
    if step_y == 1:
        y.append(y[j] + 1 + np.random.normal())
    else:
        y.append(y[j] - 1 + np.random.normal())
        
trace1 = go.Scatter(
    x=x,
    y=y,
    mode='markers',
    name='Random Walk',
    marker=dict(
        color=[i for i in range(len(x))],
        size=8,
        colorscale='Greens',
        showscale=True
    )
)

data = [trace1]
py.iplot(data, filename='random-walk-2d')
Out[3]:

Advanced Tip

We can formally think of a 1D random walk as a point jumping along the integer number line. Let $Z_i$ be a random variable that takes on the values +1 and -1. Let this random variable represent the steps we take in the random walk in 1D (where +1 means right and -1 means left). Also, as with the above visualizations, let us assume that the probability of moving left and right is just $\frac{1}{2}$. Then, consider the sum

$$ \begin{align*} S_n = \sum_{i=0}^{n}{Z_i} \end{align*} $$

where S_n represents the point that the random walk ends up on after n steps have been taken.

To find the expected value of $S_n$, we can compute it directly. Since each $Z_i$ is independent, we have

$$ \begin{align*} \mathbb{E}(S_n) = \sum_{i=0}^{n}{\mathbb{E}(Z_i)} \end{align*} $$

but since $Z_i$ takes on the values +1 and -1 then

$$ \begin{align*} \mathbb{E}(Z_i) = 1 \cdot P(Z_i=1) + -1 \cdot P(Z_i=-1) = \frac{1}{2} - \frac{1}{2} = 0 \end{align*} $$

Therefore, we expect our random walk to hover around $0$ regardless of how many steps we take in our walk.

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